Finite Group Databasebeta

← C3×#10590258 #13157311

Subgroups — C3×#10590258 #13157311

Order 1728.

558 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×C4×Dic6C211yesmaximal, characteristic
2C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
3C32×C4×C3⋊C8C211yesmaximal, characteristic
4C3×Dic3⋊(C3⋊C8)C212yesmaximal
5C3×#63554C212yesmaximal
6C3×#5024—31nomaximal
7C3×#4679—31nomaximal
8#10590258C311yesmaximal, characteristic, direct product factor
9C2×C32×Dic6C411yescharacteristic
10C33×C42C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2212yes—
12C32×C4⋊Dic3C411yescharacteristic
13C32×Dic3⋊C4C412yes—
14C32×C4×Dic3C2212yes—
15C2×C32×C3⋊C8C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17C32×C4×Q8—31no—
18C3×C4×Dic6—21no—
19C3×C4×Dic6D311yescharacteristic
20C3×C4×Dic6C2×C311yescharacteristic
21C32×C4×C8—31no—
22C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic
23C3×#815—32no—
24C3×Dic3⋊C8—32no—
25C3×C4⋊(C3⋊C8)—31no—
26C3×C4⋊(C3⋊C8)—31no—
27C3×C4⋊(C3⋊C8)—32no—
28C3×C4⋊(C3⋊C8)—61no—
29C3×#803—32no—
30C3×C4×C3⋊C8—21no—
31C3×C4×C3⋊C8D311yescharacteristic
32C3×C4×C3⋊C8C2×C311yescharacteristic
33Dic3⋊(C3⋊C8)C2×C312yes—
34#63554C2×C312yes—
35C2×C33×C4C2×C411yescharacteristic
36C2×C33×C4C2×C411yescharacteristic
37C2×C33×C4C2311yescharacteristic
38C2×C32×Dic3C2×C412yes—
39C2×C32×Dic3C2×C412yes—
40C32×Dic6—22no—
41C3×C3⋊(C3⋊C8)—22no—
42C32×C3⋊C8—21no—
43C32×C3⋊C8Q812yescomplemented
44C3×#446—91no—
45#5024—31no—
46#4679—31no—
47C2×C32×Q8—31no—
48C2×C3×Dic6—21no—
49C2×C3×Dic6Dic311yescharacteristic
50C2×C3×Dic6C3×C411yescharacteristic
51C2×C32×C8—31no—
52C2×C32×C8—31no—
53C32×C4⋊C4—32no—
54C32×C4⋊C4—31no—
55C32×C42—21no—
56C32×C42—21no—
57C32×C42—21no—
58C32×C42—32no—
59C32×C42—41no—
60C32×C42C2×D311yescharacteristic
61C32×C42C2×D311yescharacteristic
62C32×C42C22×C311yescharacteristic
63C2×C3⋊(C3⋊C8)C22×C312yes—
64C3×C4⋊Dic3—21no—
65C3×C4⋊Dic3Dic311yescharacteristic
66C3×C4⋊Dic3C3×C411yescharacteristic
67C3×Dic3⋊C4—22no—
68C3×Dic3⋊C4Dic312yes—
69C3×Dic3⋊C4C3×C412yes—
70C3×C4×Dic3—22no—
71C3×C4×Dic3C2×D312yes—
72C3×C4×Dic3C22×C312yes—
73C2×C3×C3⋊C8—21no—
74C2×C3×C3⋊C8—21no—
75C2×C3×C3⋊C8—32no—
76C2×C3×C3⋊C8—32no—
77C2×C3×C3⋊C8—62no—
78C2×C3×C3⋊C8C2×D311yescharacteristic
79C2×C3×C3⋊C8C2×D311yescharacteristic
80C2×C3×C3⋊C8C22×C311yescharacteristic
81C2×C3×C3⋊C8C22×C311yescharacteristic
82C22×C33C22×C411yescharacteristic
83C33×C4—21no—
84C33×C4M1612yes—
85C33×C4C2×Q811yescharacteristic
86C33×C4#4111yescharacteristic
87C32×Dic3—22no—
88C32×Dic3—22no—
89C3×C4×Q8—31no—
90C3×C4×Q8—31no—
91C3×C4×Q8—61no—
92C4×Dic6C3×D311yescharacteristic
93C3×C4⋊C8—92no—
94C3×C4⋊C8—91no—
95C3×#97—92no—
96C3×C4×C8—31no—
97C3×C4×C8—31no—
98C3×C4×C8—61no—
99#815—32no—
100Dic3⋊C8—32no—
101C4⋊(C3⋊C8)—31no—
102C4⋊(C3⋊C8)—31no—
103C4⋊(C3⋊C8)—32no—
104C4⋊(C3⋊C8)—61no—
105#803—32no—
106C4×C3⋊C8C3×D311yescharacteristic
107C32×Q8—62no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—32no—
118C2×C32×C4—32no—
119C2×C32×C4—41no—
120C2×C32×C4—41no—
121C2×C32×C4—41no—
122C2×C32×C4C4×D311yescharacteristic
123C2×C32×C4C4×D311yescharacteristic
124C2×C32×C4C2×Dic311yescharacteristic
125C2×C32×C4C2×Dic311yescharacteristic
126C2×C32×C4C2×C3×C411yescharacteristic
127C2×C32×C4C2×C3×C411yescharacteristic
128C2×C32×C4C22×D311yescharacteristic
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C23×C311yescharacteristic
131C2×C3×Dic3—22no—
132C2×C3×Dic3—22no—
133C2×C3×Dic3C2×Dic312yes—
134C2×C3×Dic3C2×Dic312yes—
135C2×C3×Dic3C2×C3×C412yes—
136C2×C3×Dic3C2×C3×C412yes—
137C3×Dic6—24no—
138C3×Dic6—22no—
139C3×Dic6—22no—
140C32×C8—32no—
141C32×C8—61no—
142C3⋊(C3⋊C8)—22no—
143C3×C3⋊C8—22no—
144C3×C3⋊C8—21no—
145C3×C3⋊C8—21no—
146C3×C3⋊C8—22no—
147C3×C3⋊C8—62no—
148C3×C3⋊C8—62no—
149C3×C3⋊C8—122no—
150C3×C3⋊C8Dic612yescomplemented
151C3×C3⋊C8C3×Q812yescomplemented
152#446—91noSylow
153C2×C33C4×Q811yescharacteristic
154C2×C33C2×M1611yescharacteristic
155C2×C33#13111yescharacteristic
156C2×C3×Q8—31no—
157C2×C3×Q8—31no—
158C2×C3×Q8—61no—
159C2×Dic6C3×Dic311yescharacteristic
160C2×C3×C8—31no—
161C2×C3×C8—31no—
162C2×C3×C8—31no—
163C2×C3×C8—31no—
164C2×C3×C8—61no—
165C2×C3×C8—61no—
166C2×C3×C8—92no—
167C3×C4⋊C4—32no—
168C3×C4⋊C4—31no—
169C3×C4⋊C4—32no—
170C3×C4⋊C4—31no—
171C3×C4⋊C4—62no—
172C3×C4⋊C4—61no—
173C3×C42—21no—
174C3×C42—21no—
175C3×C42—21no—
176C3×C42—32no—
177C3×C42—32no—
178C3×C42—41no—
179C3×C42—62no—
180C3×C42D3211yescharacteristic
181C3×C42C2×C3×D311yescharacteristic
182C3×C42C2×C3×D311yescharacteristic
183C4⋊Dic3C3×Dic311yescharacteristic
184Dic3⋊C4C3×Dic312yes—
185C4×Dic3C2×C3×D312yes—
186C2×C3⋊C8—32no—
187C2×C3⋊C8—32no—
188C2×C3⋊C8—62no—
189C2×C3⋊C8C2×C3×D311yescharacteristic
190C2×C3⋊C8C2×C3×D311yescharacteristic
191C22×C32—21no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—41no—
195C22×C32C2×C4×D311yescharacteristic
196C22×C32C22×Dic311yescharacteristic
197C22×C32C22×C3×C411yescharacteristic
198C32×C4—21no—
199C32×C4—22no—
200C32×C4—21no—
201C32×C4—21no—
202C32×C4—22no—
203C32×C4—22no—
204C32×C4—22no—
205C32×C4—21no—
206C32×C4—21no—
207C32×C4—21no—
208C32×C4—21no—
209C32×C4—21no—
210C32×C4—21no—
211C32×C4—42no—
212C32×C4—41no—
213C32×C4—41no—
214C32×C4—41no—
215C32×C4—41no—
216C32×C4—42no—
217C32×C4—62no—
218C32×C4—62no—
219C32×C4ΓL(1,𝔽25)12yes—
220C32×C4#20812yes—
221C32×C4C3×M1612yes—
222C32×C4C2×Dic611yescharacteristic
223C32×C4#23511yescharacteristic
224C32×C4D3×Q811yescharacteristic
225C32×C4#23911yescharacteristic
226C32×C4C2×C3×Q811yescharacteristic
227C32×C4C3×#4111yescharacteristic
228C3×Dic3—24no—
229C3×Dic3—22no—
230C3×Dic3—22no—
231C3×Dic3—22no—
232C3×Dic3—22no—
233C3×Dic3—42no—
234C4×Q8—31no—
235C4⋊C8—92no—
236C4⋊C8—91no—
237#97—92no—
238C4×C8—31no—
239C33#44611yescharacteristic, complemented, Sylow
240C3×Q8—64no—
241C3×Q8—62no—
242C3×Q8—62no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—32no—
253C2×C3×C4—32no—
254C2×C3×C4—32no—
255C2×C3×C4—32no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—62no—
260C2×C3×C4—62no—
261C2×C3×C4D3×Dic311yescharacteristic
262C2×C3×C4D3×Dic311yescharacteristic
263C2×C3×C4C3×C4×D311yescharacteristic
264C2×C3×C4C3×C4×D311yescharacteristic
265C2×C3×C4C2×C3×Dic311yescharacteristic
266C2×C3×C4C2×C3×Dic311yescharacteristic
267C2×C3×C4C2×D3211yescharacteristic
268C2×C3×C4C22×C3×D311yescharacteristic
269C2×C3×C4C22×C3×D311yescharacteristic
270C2×Dic3C2×C3×Dic312yes—
271C2×Dic3C2×C3×Dic312yes—
272Dic6—22no—
273C3×C8—32no—
274C3×C8—32no—
275C3×C8—62no—
276C3×C8—62no—
277C3×C8—61no—
278C3×C8—61no—
279C3×C8—182no—
280C3⋊C8—21no—
281C3⋊C8—62no—
282C3⋊C8—62no—
283C3⋊C8—122no—
284C3⋊C8C3×Dic612yescomplemented
285C2×C32—21no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32—21no—
290C2×C32—21no—
291C2×C32—21no—
292C2×C32—21no—
293C2×C32—21no—
294C2×C32—41no—
295C2×C32—41no—
296C2×C32—41no—
297C2×C32C4×Dic611yescharacteristic
298C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
299C2×C32#90111yescharacteristic
300C2×C32C2×#20811yescharacteristic
301C2×C32Q8×Dic311yescharacteristic
302C2×C32#94811yescharacteristic
303C2×C32C3×C4×Q811yescharacteristic
304C2×C32C2×C3×M1611yescharacteristic
305C2×C32C3×#13111yescharacteristic
306C2×Q8—31no—
307C2×C8—31no—
308C2×C8—31no—
309C2×C8—92no—
310C4⋊C4—32no—
311C4⋊C4—31no—
312C42—32no—
313C42C3×D3211yescharacteristic
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—41no—
318C22×C3C2×D3×Dic311yescharacteristic
319C22×C3C2×C3×C4×D311yescharacteristic
320C22×C3C22×C3×Dic311yescharacteristic
321C3×C4—22no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—22no—
332C3×C4—22no—
333C3×C4—22no—
334C3×C4—41no—
335C3×C4—41no—
336C3×C4—42no—
337C3×C4—41no—
338C3×C4—41no—
339C3×C4—42no—
340C3×C4—64no—
341C3×C4—62no—
342C3×C4—62no—
343C3×C4—62no—
344C3×C4—62no—
345C3×C4—122no—
346C3×C4#371112yes—
347C3×C4C3×ΓL(1,𝔽25)12yes—
348C3×C4C3×#20812yes—
349C3×C4D3×Dic611yescharacteristic
350C3×C4#379911yescharacteristic
351C3×C4C2×C3×Dic611yescharacteristic
352C3×C4C3×#23511yescharacteristic
353C3×C4C3×D3×Q811yescharacteristic
354C3×C4C3×#23911yescharacteristic
355Dic3—22no—
356Dic3—22no—
357C32—21no—
358C32—21no—
359C32—21no—
360C32—41no—
361C32#467911yescharacteristic, complemented
362C32#502411yescharacteristic, complemented
363C32C3×#44611yescharacteristic, complemented
364Q8—62no—
365C2×C4—32no—
366C2×C4—32no—
367C2×C4C3×D3×Dic311yescharacteristic
368C2×C4C3×D3×Dic311yescharacteristic
369C2×C4C2×C3×D3211yescharacteristic
370C8—32no—
371C8—61no—
372C8—182no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—41no—
383C2×C3—41no—
384C2×C3—41no—
385C2×C3#6381411yescharacteristic
386C2×C3C2×#371111yescharacteristic
387C2×C3Dic3×Dic611yescharacteristic
388C2×C3C3×C4×Dic611yescharacteristic
389C2×C3C2×C3×ΓL(1,𝔽25)11yescharacteristic
390C2×C3C3×#90111yescharacteristic
391C2×C3C2×C3×#20811yescharacteristic
392C2×C3C3×Q8×Dic311yescharacteristic
393C2×C3C3×#94811yescharacteristic
394C22C2×C3×D3×Dic311yescharacteristic
395C4—21no—
396C4—62no—
397C4—62no—
398C4C3×#371112yes—
399C4C3×D3×Dic611yescharacteristic
400C4C3×#379911yescharacteristic
401C3—21no—
402C3—21no—
403C3—21no—
404C3—41no—
405C3#1059025811yescharacteristic, direct product factor
406C3C3×#467911yescharacteristic, complemented
407C3C3×#502411yescharacteristic, complemented
408C2C3×#6381411yescharacteristic
409C2C2×C3×#371111yescharacteristic
410C2C3×Dic3×Dic611yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socle, prime radicalC22×C33C22×C4
cosocleC2×C32×C4C23×C3
hypercenterC3×C42D32
hypocenterC32C3×#446
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×C42

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
8. #10590258C3direct (N × H)
43. C32×C3⋊C8Q8semidirect
150. C3×C3⋊C8Dic6semidirect
151. C3×C3⋊C8C3×Q8semidirect
239. C33#446semidirect
284. C3⋊C8C3×Dic6semidirect
361. C32#4679semidirect
362. C32#5024semidirect
363. C32C3×#446semidirect
405. C3#10590258direct (N × H)
406. C3C3×#4679semidirect
407. C3C3×#5024semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C32×C4×Dic62
C3×C4⋊(C3⋊(C3⋊C8))2
C32×C4×C3⋊C82
C3×Dic3⋊(C3⋊C8)2
C3×#635542
C3×#50243
C3×#46793
#105902583